Linking the Singularities of Cosmological Correlators

TL;DR

This paper develops a cohesive bootstrap framework for cosmological correlators whose tree-level structure is rational in the energies. By combining unitarity cutting rules, recursion relations (gluing), and a lifting/transmutation map from flat-space correlators to de Sitter space, the authors show how many cosmological four-point functions, including those with gravitons and gauge fields, can be reconstructed from simpler building blocks. The methods are demonstrated via explicit constructions such as graviton Compton scattering, illustrating how singularity data suffices to fix full correlators. The approach clarifies how de Sitter physics encodes bulk dynamics without explicit time evolution, and points toward extensions to loop-level cosmology and broader amplitude-bootstrapping strategies.

Abstract

Much of the structure of cosmological correlators is controlled by their singularities, which in turn are fixed in terms of flat-space scattering amplitudes. An important challenge is to interpolate between the singular limits to determine the full correlators at arbitrary kinematics. This is particularly relevant because the singularities of correlators are not directly observable, but can only be accessed by analytic continuation. In this paper, we study rational correlators, including those of gauge fields, gravitons, and the inflaton, whose only singularities at tree level are poles and whose behavior away from these poles is strongly constrained by unitarity and locality. We describe how unitarity translates into a set of cutting rules that consistent correlators must satisfy, and explain how this can be used to bootstrap correlators given information about their singularities. We also derive recursion relations that allow the iterative construction of more complicated correlators from simpler building blocks. In flat space, all energy singularities are simple poles, so that the combination of unitarity constraints and recursion relations provides an efficient way to bootstrap the full correlators. In many cases, these flat-space correlators can then be transformed into their more complex de Sitter counterparts. As an example of this procedure, we derive the correlator associated to graviton Compton scattering in de Sitter space, though the methods are much more widely applicable.

Figures (6)

• Figure 1: A one-dimensional slice through a four-point correlation function. Singularities in the unphysical region (where some of the energies have been analytically continued to negative values) determine the form of the correlator in the physical region.
• Figure 2: Schematic of the contour deformation used to derive \ref{['eq:residue']}. The solid contour centered around $z=0$, whose residue is the original wavefunction coefficient, $\psi(0)$, can be deformed into the dashed contour to write the wavefunction as a sum over the residues of the other poles of $\psi(z)$, located at $z_j$.
• Figure 3: Illustration of the different contributions to the Abelian Compton correlator. Consistency requires the exchange of particles in both the $s$ and $t$-channels, along with a particular contact contribution. In the recursive construction of the correlator, these contributions are linked together by the total energy singularity.
• Figure 4: Illustration of the transmutation operators for $\langle J_\ell\varphi\varphi\varphi\rangle$ ( left) and $\langle \varphi \varphi \varphi \varphi \rangle^{(\ell)}$ ( right).
• Figure 5: Illustration of the different contributions to gravitational Compton scattering.